# zbMATH — the first resource for mathematics

Countably compact and pseudocompact topologies on free Abelian groups. (English. Russian original) Zbl 0714.22001
Sov. Math. 34, No. 5, 79-86 (1990); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 5(336), 68-75 (1990).
It is well known that any non-trivial free Abelian group does not admit a compact Hausdorff group topology. In connection with this result the author proves the following results: Theorem 1 (CH). On the free Abelian group G of cardinality $$c=2^{\aleph_ 0}$$ there exists a countably compact Hausdorff group topology $${\mathcal T}$$ such that the group $$<G,{\mathcal T}>$$ is connected, locally connected, hereditarily separable and hereditarily normal. Theorem 2. On the free Abelian group of cardinality $$2^{\aleph_ 0}$$ there exists a chain $${\mathfrak A}$$ of pseudocompact Hausdorff group topologies such that $$| {\mathfrak A}| =2^{(c^+)}$$, where $$c^+$$ is the smallest cardinal larger than c.
Reviewer: M.I.Ursul

##### MSC:
 22A05 Structure of general topological groups 20K45 Topological methods for abelian groups 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 20K20 Torsion-free groups, infinite rank